OR Probability Calculator

About the OR Probability Calculator

The OR probability calculator computes P(A or B) — the probability that at least one of two (or three) events occurs. It supports three modes: mutually exclusive events (P(A∪B) = P(A) + P(B)), non-exclusive events (P(A∪B) = P(A) + P(B) − P(A∩B)), and three-event inclusion-exclusion.

Understanding OR probability is essential for risk assessment ("probability of failure A or failure B"), insurance ("probability of claim type A or B"), and everyday probability reasoning. The addition rule is one of the most commonly needed probability formulas.

This calculator visualizes the probability breakdown with stacked bars, compares all probability components side-by-side, and shows how the probability of "at least one" occurrence scales across repeated independent trials. It is a fast way to see whether you need to subtract overlap, use a complement, or switch to inclusion-exclusion for multiple events.

When This Page Helps

The addition rule is one of the most frequently needed probability calculations, from simple "what are the odds of X or Y?" questions to larger risk models with multiple failure modes. This calculator helps you avoid double-counting, compare the overlap against the union, and confirm that the parts still sum to the whole.

It is useful whenever you need a clean answer to "A or B", whether that means a coin result, a failure condition, or a compound event in a probability table.

How to Use the Inputs

1. Select the mode: mutually exclusive, non-exclusive, or three events.
2. Enter P(A) and P(B) as percentages.
3. For non-exclusive mode, enter P(A ∩ B) — the joint probability.
4. For three events, enter P(C) as well (assumes independence).
5. Set the number of trials for the repeated-trial analysis.
6. Review the probability comparison table and stacked bar visualization.

Example Calculation

Tips & Best Practices

• Mutually exclusive events cannot both occur — think of a single die roll being 1 OR 6 (can't be both). Here P(A∩B) = 0.
• Non-exclusive events: rolling an even number OR a number > 4. Rolling 6 satisfies both, so subtract the overlap.
• The complement approach is often easier: P(A or B) = 1 − P(neither) = 1 − P(A')×P(B') for independent events.
• In the repeated trials table, "at least one" grows quickly — 10 trials at 30% per trial gives 97.2% chance of at least one occurrence.
• For non-independent events, you must know P(A∩B) separately — it's not simply P(A)×P(B).
• The stacked bar gives an instant visual of how the total probability space is divided.

Inclusion-Exclusion for Multiple Events

For n events, the inclusion-exclusion formula alternates adding and subtracting intersections: P(A₁∪...∪Aₙ) = ΣP(Aᵢ) − ΣP(Aᵢ∩Aⱼ) + ΣP(Aᵢ∩Aⱼ∩Aₖ) − ... This formula has 2ⁿ−1 terms, making it impractical for many events. In such cases, the complement method is preferred.

Applications in Reliability Engineering

System reliability often uses OR probability. A system fails if component A OR component B fails. For independent components with failure probabilities pA and pB, system failure probability = pA + pB − pA×pB. Redundant systems reverse this: the system survives if component A OR component B survives.

De Morgan's Laws and Complements

De Morgan's laws connect OR and AND through complements: P(A∪B) = 1 − P(A'∩B') and P(A∩B) = 1 − P(A'∪B'). These identities are fundamental for converting between "at least one" and "all/none" probability problems.

Frequently Asked Questions

• What's the difference between OR and AND probability?

  OR (union) asks "at least one event occurs" and is P(A) + P(B) − P(A∩B). AND (intersection) asks "both events occur" and is P(A)×P(B) for independent events. OR always gives a larger or equal probability.

• When are events mutually exclusive?

  When they cannot happen simultaneously. A coin landing heads AND tails is impossible. A single card being both a heart and a spade is impossible. For mutually exclusive events, P(A∩B) = 0.

• What does the stacked bar show?

  It divides the total probability space into: only A, both A and B, only B, and neither. These four regions always sum to 100%.

• How does the three-event mode work?

  It uses the inclusion-exclusion principle for three events. For independent events, all pairwise and triple intersections are computed from the marginal probabilities.

• Why does "at least one in n trials" grow so fast?

  Each trial has probability (1−P(A∪B)) of "no occurrence." After n trials, P(none) = (1−P(A∪B))^n, which shrinks exponentially. So P(at least one) = 1 − (1−P(A∪B))^n grows rapidly.

• Can P(A or B) exceed 100%?

  No. P(A or B) is capped at 100%. If P(A) + P(B) > 100% and they're "mutually exclusive," the probabilities are inconsistent — genuinely mutually exclusive events can't have P(A) + P(B) > 1.